feat: Refactor Index notation
This commit is contained in:
jstoobysmith
parent
3f5eb58db4
commit
809b80ff88
5 changed files with 195 additions and 37 deletions
|
|
@ -91,7 +91,6 @@ import HepLean.SpaceTime.LorentzVector.NormOne
|
|||
import HepLean.SpaceTime.MinkowskiMetric
|
||||
import HepLean.SpaceTime.SL2C.Basic
|
||||
import HepLean.SpaceTime.WeylFermion.Basic
|
||||
import HepLean.SpaceTime.WeylFermion.ColorFun
|
||||
import HepLean.SpaceTime.WeylFermion.Modules
|
||||
import HepLean.StandardModel.Basic
|
||||
import HepLean.StandardModel.HiggsBoson.Basic
|
||||
|
|
@ -100,7 +99,6 @@ import HepLean.StandardModel.HiggsBoson.PointwiseInnerProd
|
|||
import HepLean.StandardModel.HiggsBoson.Potential
|
||||
import HepLean.StandardModel.Representations
|
||||
import HepLean.Tensors.Basic
|
||||
import HepLean.Tensors.ColorCat.Basic
|
||||
import HepLean.Tensors.Contraction
|
||||
import HepLean.Tensors.EinsteinNotation.Basic
|
||||
import HepLean.Tensors.EinsteinNotation.IndexNotation
|
||||
|
|
|
|||
61
HepLean/Tensors/ComplexLorentz/Basic.lean
Normal file
61
HepLean/Tensors/ComplexLorentz/Basic.lean
Normal file
|
|
@ -0,0 +1,61 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.Tensors.OverColor.Basic
|
||||
import HepLean.Mathematics.PiTensorProduct
|
||||
/-!
|
||||
|
||||
## Complex Lorentz tensors
|
||||
|
||||
-/
|
||||
|
||||
namespace Fermion
|
||||
|
||||
open Matrix
|
||||
open MatrixGroups
|
||||
open Complex
|
||||
open TensorProduct
|
||||
open IndexNotation
|
||||
open CategoryTheory
|
||||
open MonoidalCategory
|
||||
|
||||
/-- The colors associated with complex representations of SL(2, ℂ) of intrest to physics. -/
|
||||
inductive Color
|
||||
| upL : Color
|
||||
| downL : Color
|
||||
| upR : Color
|
||||
| downR : Color
|
||||
| up : Color
|
||||
| down : Color
|
||||
|
||||
def τ : Color → Color
|
||||
| Color.upL => Color.downL
|
||||
| Color.downL => Color.upL
|
||||
| Color.upR => Color.downR
|
||||
| Color.downR => Color.upR
|
||||
| Color.up => Color.down
|
||||
| Color.down => Color.up
|
||||
|
||||
def evalNo : Color → ℕ
|
||||
| Color.upL => 2
|
||||
| Color.downL => 2
|
||||
| Color.upR => 2
|
||||
| Color.downR => 2
|
||||
| Color.up => 4
|
||||
| Color.down => 4
|
||||
|
||||
noncomputable section
|
||||
/-- The corresponding representations associated with a color. -/
|
||||
def colorToRep (c : Color) : Rep ℂ SL(2, ℂ) :=
|
||||
match c with
|
||||
| Color.upL => Fermion.altLeftHanded
|
||||
| Color.downL => Fermion.leftHanded
|
||||
| Color.upR => Fermion.altRightHanded
|
||||
| Color.downR => Fermion.rightHanded
|
||||
| Color.up => Lorentz.complexContr
|
||||
| Color.down => Lorentz.complexCo
|
||||
|
||||
end
|
||||
end Fermion
|
||||
|
|
@ -3,6 +3,7 @@ Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
|||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.Tensors.ComplexLorentz.Basic
|
||||
import HepLean.Tensors.OverColor.Basic
|
||||
import HepLean.Mathematics.PiTensorProduct
|
||||
/-!
|
||||
|
|
@ -21,40 +22,6 @@ open TensorProduct
|
|||
open IndexNotation
|
||||
open CategoryTheory
|
||||
|
||||
/-- The colors associated with complex representations of SL(2, ℂ) of intrest to physics. -/
|
||||
inductive Color
|
||||
| upL : Color
|
||||
| downL : Color
|
||||
| upR : Color
|
||||
| downR : Color
|
||||
| up : Color
|
||||
| down : Color
|
||||
|
||||
def τ : Color → Color
|
||||
| Color.upL => Color.downL
|
||||
| Color.downL => Color.upL
|
||||
| Color.upR => Color.downR
|
||||
| Color.downR => Color.upR
|
||||
| Color.up => Color.down
|
||||
| Color.down => Color.up
|
||||
|
||||
def evalNo : Color → ℕ
|
||||
| Color.upL => 2
|
||||
| Color.downL => 2
|
||||
| Color.upR => 2
|
||||
| Color.downR => 2
|
||||
| Color.up => 4
|
||||
| Color.down => 4
|
||||
/-- The corresponding representations associated with a color. -/
|
||||
def colorToRep (c : Color) : Rep ℂ SL(2, ℂ) :=
|
||||
match c with
|
||||
| Color.upL => altLeftHanded
|
||||
| Color.downL => leftHanded
|
||||
| Color.upR => altRightHanded
|
||||
| Color.downR => rightHanded
|
||||
| Color.up => Lorentz.complexContr
|
||||
| Color.down => Lorentz.complexCo
|
||||
|
||||
/-- The linear equivalence between `colorToRep c1` and `colorToRep c2` when `c1 = c2`. -/
|
||||
def colorToRepCongr {c1 c2 : Color} (h : c1 = c2) : colorToRep c1 ≃ₗ[ℂ] colorToRep c2 where
|
||||
toFun := Equiv.cast (congrArg (CoeSort.coe ∘ colorToRep) h)
|
||||
|
|
@ -192,6 +159,12 @@ lemma map_tprod {X Y : OverColor Color} (p : (i : X.left) → (colorToRep (X.hom
|
|||
((PiTensorProduct.tprod ℂ) p)) = _
|
||||
rw [PiTensorProduct.reindex_tprod, PiTensorProduct.congr_tprod]
|
||||
|
||||
lemma obj_ρ_tprod (f : OverColor Color) (M : SL(2, ℂ))
|
||||
(x : (i : f.left) → CoeSort.coe (colorToRep (f.hom i))) :
|
||||
(colorFun.obj f).ρ M ((PiTensorProduct.tprod ℂ) x) =
|
||||
PiTensorProduct.tprod ℂ (fun i => (colorToRep (f.hom i)).ρ M (x i)) := by
|
||||
exact obj'_ρ_tprod _ _ _
|
||||
|
||||
@[simp]
|
||||
lemma obj_ρ_empty (g : SL(2, ℂ)) : (colorFun.obj (𝟙_ (OverColor Color))).ρ g = LinearMap.id := by
|
||||
erw [colorFun.obj'_ρ]
|
||||
|
|
|
|||
|
|
@ -0,0 +1,123 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.Tensors.OverColor.Basic
|
||||
import HepLean.Tensors.OverColor.Functors
|
||||
import HepLean.Tensors.COmplexLorentz.ColorFun
|
||||
import HepLean.Mathematics.PiTensorProduct
|
||||
/-!
|
||||
|
||||
## The contraction monoidal natural transformation
|
||||
|
||||
-/
|
||||
|
||||
namespace Fermion
|
||||
|
||||
noncomputable section
|
||||
|
||||
open Matrix
|
||||
open MatrixGroups
|
||||
open Complex
|
||||
open TensorProduct
|
||||
open IndexNotation
|
||||
open CategoryTheory
|
||||
open MonoidalCategory
|
||||
|
||||
def tensorateContrPair (c : OverColor Color) : (OverColor.contrPair Color τ ⊗⋙ colorFunMon).obj c ≅
|
||||
(colorFunMon.obj c) ⊗ colorFunMon.obj ((OverColor.map τ).obj c) :=
|
||||
(colorFunMon.μIso c ((OverColor.map τ).obj c)).symm
|
||||
|
||||
namespace pairwiseRepFun
|
||||
|
||||
def obj' (c : OverColor Color) : Rep ℂ SL(2, ℂ) := Rep.of {
|
||||
toFun := fun M => PiTensorProduct.map (fun x =>
|
||||
TensorProduct.map ((colorToRep (c.hom x)).ρ M) ((colorToRep (τ (c.hom x))).ρ M)),
|
||||
map_one' := by
|
||||
simp
|
||||
map_mul' := fun x y => by
|
||||
simp only [Functor.id_obj, _root_.map_mul]
|
||||
ext x' : 2
|
||||
simp only [LinearMap.compMultilinearMap_apply, PiTensorProduct.map_tprod, LinearMap.mul_apply]
|
||||
apply congrArg
|
||||
funext i
|
||||
change _ = (TensorProduct.map _ _ ∘ₗ TensorProduct.map _ _ ) (x' i)
|
||||
rw [← TensorProduct.map_comp]
|
||||
rfl}
|
||||
|
||||
/-- Given a morphism in `OverColor Color` the corresopnding linear equivalence between `obj' _`
|
||||
induced by reindexing. -/
|
||||
def mapToLinearEquiv' {f g : OverColor Color} (m : f ⟶ g) : (obj' f).V ≃ₗ[ℂ] (obj' g).V :=
|
||||
(PiTensorProduct.reindex ℂ (fun x => (colorToRep (f.hom x)).V ⊗[ℂ] (colorToRep (τ (f.hom x))).V)
|
||||
(OverColor.Hom.toEquiv m)).trans
|
||||
(PiTensorProduct.congr (fun i =>
|
||||
TensorProduct.congr (colorToRepCongr (OverColor.Hom.toEquiv_symm_apply m i))
|
||||
((colorToRepCongr (congrArg τ (OverColor.Hom.toEquiv_symm_apply m i))))))
|
||||
|
||||
lemma mapToLinearEquiv'_tprod {f g : OverColor Color} (m : f ⟶ g)
|
||||
(x : (i : f.left) → (colorToRep (f.hom i)).V ⊗[ℂ] (colorToRep (τ (f.hom i))).V) :
|
||||
mapToLinearEquiv' m (PiTensorProduct.tprod ℂ x) =
|
||||
PiTensorProduct.tprod ℂ fun i =>
|
||||
(TensorProduct.congr (colorToRepCongr (OverColor.Hom.toEquiv_symm_apply m i ))
|
||||
(colorToRepCongr (mapToLinearEquiv'.proof_4 m i ))) (x ((OverColor.Hom.toEquiv m).symm i)) := by
|
||||
simp [mapToLinearEquiv']
|
||||
change (PiTensorProduct.congr fun i => TensorProduct.congr (colorToRepCongr _) (colorToRepCongr _))
|
||||
((PiTensorProduct.reindex ℂ (fun x => ↑(colorToRep (f.hom x)).V ⊗[ℂ] ↑(colorToRep (τ (f.hom x))).V)
|
||||
(OverColor.Hom.toEquiv m))
|
||||
((PiTensorProduct.tprod ℂ) x)) = _
|
||||
rw [PiTensorProduct.reindex_tprod]
|
||||
erw [PiTensorProduct.congr_tprod]
|
||||
rfl
|
||||
|
||||
|
||||
end pairwiseRepFun
|
||||
|
||||
def pairwiseRep (c : OverColor Color) : Rep ℂ SL(2, ℂ) := Rep.of {
|
||||
toFun := fun M => PiTensorProduct.map (fun x =>
|
||||
TensorProduct.map ((colorToRep (c.hom x)).ρ M) ((colorToRep (τ (c.hom x))).ρ M )),
|
||||
map_one' := by
|
||||
simp
|
||||
map_mul' := fun x y => by
|
||||
simp only [Functor.id_obj, _root_.map_mul]
|
||||
ext x' : 2
|
||||
simp only [LinearMap.compMultilinearMap_apply, PiTensorProduct.map_tprod, LinearMap.mul_apply]
|
||||
apply congrArg
|
||||
funext i
|
||||
change _ = (TensorProduct.map _ _ ∘ₗ TensorProduct.map _ _ ) (x' i)
|
||||
rw [← TensorProduct.map_comp]
|
||||
rfl}
|
||||
|
||||
def contrPairPairwiseRep (c : OverColor Color) :
|
||||
(colorFunMon.obj c) ⊗ colorFunMon.obj ((OverColor.map τ).obj c) ⟶
|
||||
pairwiseRep c where
|
||||
hom := TensorProduct.lift (PiTensorProduct.map₂ (fun x =>
|
||||
TensorProduct.mk ℂ (colorToRep (c.hom x)).V (colorToRep (τ (c.hom x))).V))
|
||||
comm M := by
|
||||
refine HepLean.PiTensorProduct.induction_tmul (fun x y => ?_)
|
||||
simp only [Functor.id_obj, CategoryStruct.comp, Action.instMonoidalCategory_tensorObj_V,
|
||||
Equivalence.symm_inverse, Action.functorCategoryEquivalence_functor,
|
||||
Action.FunctorCategoryEquivalence.functor_obj_obj, Action.tensor_ρ', LinearMap.coe_comp,
|
||||
Function.comp_apply]
|
||||
change (TensorProduct.lift
|
||||
(PiTensorProduct.map₂ fun x => TensorProduct.mk ℂ ↑(colorToRep (c.hom x)).V ↑(colorToRep (τ (c.hom x))).V))
|
||||
((TensorProduct.map _ _)
|
||||
((PiTensorProduct.tprod ℂ) x ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) y)) = _
|
||||
rw [TensorProduct.map_tmul]
|
||||
erw [colorFun.obj_ρ_tprod, colorFun.obj_ρ_tprod]
|
||||
simp only [Functor.id_obj, lift.tmul]
|
||||
erw [PiTensorProduct.map₂_tprod_tprod]
|
||||
change _ = ((pairwiseRep c).ρ M)
|
||||
((TensorProduct.lift
|
||||
(PiTensorProduct.map₂ fun x => TensorProduct.mk ℂ ↑(colorToRep (c.hom x)).V ↑(colorToRep (τ (c.hom x))).V))
|
||||
((PiTensorProduct.tprod ℂ) x ⊗ₜ[ℂ] (PiTensorProduct.tprod ℂ) y))
|
||||
simp only [mk_apply, Functor.id_obj, lift.tmul]
|
||||
rw [PiTensorProduct.map₂_tprod_tprod]
|
||||
simp only [pairwiseRep, Functor.id_obj, Rep.coe_of, Rep.of_ρ, MonoidHom.coe_mk, OneHom.coe_mk,
|
||||
mk_apply]
|
||||
erw [PiTensorProduct.map_tprod]
|
||||
rfl
|
||||
|
||||
|
||||
end
|
||||
end Fermion
|
||||
|
|
@ -32,7 +32,10 @@ def upDown : Fin 2 → complexLorentzTensor.C
|
|||
|
||||
variable (T S : complexLorentzTensor.F.obj (OverColor.mk upDown))
|
||||
|
||||
#check {T | i i}ᵀ.dot
|
||||
/-
|
||||
#check {T | i m ⊗ S | m l}ᵀ.dot
|
||||
#check {T | i m ⊗ S | l τ(m)}ᵀ.dot
|
||||
-/
|
||||
end complexLorentzTensor
|
||||
|
||||
end
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue